Earlier date for Malayalam

By around 800 CE, the language used in Kerala was a local variation of Tamil. The language known as Malayalam did not exist. Ilango Adigal (who wrote Silappatikaram) and Kulasekhara Alwar (responsible for devotional literature) were Malayalis, but wrote in Tamil. By around the 9th century CE, Malayalam evolved into a separate language under the heavy influence of Saṃskṛtam. As it became a new language — the youngest among the Dravidian languages — it discarded the earlier script and started using the script used for writing Saṃskṛtam.
But some data from Edakkal caves is changing our understanding of the evolution of Malayalam language. It seems the language started evolving much earlier than the “brahminical period”.

Eminent epigraphist Iravatham Mahadevan contends that Edakal-5 provides important evidence that the common people of Kerala were already expressing themselves in Malayalam at about the end of the 4th century c.e.
The inscription (Edakal-5) is engraved just below and to the left of a tall, imposing anthropomorphic figure, which is part of the much earlier prehistoric engravings covering the rock walls of the cave (Picture 1). It appears that Edakal-5 is a label inscription engraved by a casual visitor to the cave recording his impression of the anthropomorphic figure he saw there.
The language of Edakal-5 is Malayalam. This becomes clear from the first word i (this), which is a pronoun in Malayalam standing for someone or something nearer the speaker. In Tamil, i has the same meaning, but does not occur as an independent word unlike in Malayalam. That the language of the inscription is indeed Malayalam is made clear by the second word pazhama which corresponds to pazhamai in Tamil, meaning “that which is ancient or old”. The text in Malayalam and its nearest rendering in Tamil are juxtaposed below to bring out the distinction.
i pazhama (Malayalam)
idu pazhamai (Tamil)
‘this (is) ancient’ (translation)
The most important result from the revised reading is that Edakal-5 is by far the earliest inscription in Malayalam and the only one in Brahmi. It may be assigned to late 4th or early 5th century c.e. on palaeographic evidence discussed below. The next earliest inscriptions in Malayalam occur much later from about the beginning of the 9th century c.e. and are in the Vatteluttu script.
Edakal-5 provides important evidence that the common people of Kerala were already expressing themselves in Malayalam at about the end of the 4th century c.e. However, Tamil was also retained by the elite as the literary idiom in which great works like Silappadikaram were composed. Eventually, of course, the people’s language prevailed in the region and Malayalam became the medium of communication for all purposes from about the beginning of the Kollam Era (early 9th century c.e.).[The earliest inscription in Malayalam (via Nikhil Narayanan)]

But then this is just one data point and one cannot generalize anything about what was happening in Kerala just from this.